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hep-th/9906200RU-99-25HUTP-99/A011

D-branes on the Quintic

Ilka Brunner1, Michael R. Douglas1,2,

Albion Lawrence3 and Christian Römelsberger1.

1Department of Physics and Astronomy

Rutgers University

Piscataway, NJ 08855–0849

2 I.H.E.S., Le Bois-Marie, Bures-sur-Yvette, 91440 France

3Department of Physics

Harvard University

Cambridge, MA 02138

We study D-branes on the quintic CY by combining results from several directions: general

results on holomorphic curves and vector bundles, stringy geometry and mirror symmetry,

and the boundary states in Gepner models recently constructed by Recknagel and Schome-

rus, to begin sketching a picture of D-branes in the stringy regime. We also make first

steps towards computing superpotentials on the D-brane world-volumes.

June 1999

http://arxiv.org/abs/hep-th/9906200v3

1. Introduction

In this work we study D-branes on the quintic Calabi-Yau, historically the first CY

to be intensively studied. Our guiding question will be: to classify all supersymmetry-

preserving D-branes at each point in CY moduli space, and find their world-volume moduli

spaces. As is well known, results of this type are quite relevant for phenomenological

applications of M/string theory, because the world-volume theories we will obtain include

a wide variety of four-dimensional theories with N = 1 space-time supersymmetry. Theproblem includes the classification of holomorphic vector bundles (which are ground states

for wrapped six-branes); and almost all M/string compactifications which lead to d = 4,

N = 1 supersymmetry (such as (0, 2) heterotic string compactifications and F theoryconstructions) have a choice of bundle as one of the inputs. Thus, many works have

addressed this subject explicitly or implicitly.

As usual in string compactification this geometric data is only an input and one would

really like to answer the same questions with stringy corrections included. The primary

question along these lines is: is the effect of stringy corrections just quantitative – affecting

masses and couplings in the effective Lagrangian but preserving the spectrum and moduli

spaces – or is it qualitative? If the latter, we might imagine that geometric branes undergo

radical changes of their moduli space or are even destabilized in the stringy regime, with

new branes which were unstable in the large volume limit taking their place. It should be

realized that at present very little is known about this question; for example it has not

been ruled out that the D0-brane becomes unstable in the stringy regime or has moduli

space dimension different from 3.

Clearly these questions are of great importance for the string phenomenology men-

tioned above and were asked long ago in the context of (0, 2) models. No simple answer

has been proposed; we will return to this in the conclusions.

A concrete framework which allows an exact CFT study of the stringy regime is

provided by the Gepner models. The main lesson from the original study of Gepner models

for type II and heterotic strings was that these CFT compactifications are continuously

connected to CY compactification. Mirror symmetry is manifest in the 2d superconformal

field theory, and this connection was one of the earliest arguments for it in the CY context.

The first detailed study of D-branes in Gepner models was made by Recknagel and

Schomerus [7] who (following the general approach of Cardy) constructed a large set of

examples; further work appears in [8,9]. So far no geometric interpretation or contact with

1

the large volume limit has been made. We will do so in this work. The main tool we will

use is the (symplectic) intersection form for three-cycles in the large volume limit. This

form governs Dirac quantization in the effective d = 4 theory and as such must be invariant

under any variation of the moduli. As argued in [10,11] it is given by the index Tr ab(−1)Fin open string CFT and thus is easily computed for the Gepner boundary states. The

detailed study of Kähler moduli space by Candelas et. al. [12] then allows relating this to

the large volume basis for 2p-branes. We can also make the large volume identification for

the 3-branes, aided by the large discrete symmetry group.

The detailed outline of the paper is as follows. In section 2 we review the quintic, its

homology and moduli space, and give a general overview of D-branes on the quintic in the

large volume limit. In section 3 we review the stringy geometry of its Kähler moduli space

and the monodromy group acting on B branes. In section 4 we review Gepner models

and Cardy’s theory of boundary states, which will allow us to review the boundary states

constructed by Recknagel and Schomerus. We briefly discuss the theory for K3 compacti-

fications, and show that the results agree with geometric expectations; in particular that

the dimension of a brane moduli space on K3 is given by the Mukai formula. In section

5 we compute the large volume charges for the quintic boundary states, and compute the

number of marginal operators. This will allow us to propose candidate geometric identi-

fications. In section 6 we discuss the computation of world-volume superpotentials. We

begin by presenting evidence that the superpotential is “topological” in a sense that we

explain. If true, an important consequence would be that the superpotential for B-type

branes has relatively trivial Kähler dependence and can thus be computed in the large

volume limit. This would imply general agreement between stringy and geometric results,

analogous to the case of the prepotential. In section 7 we discuss superpotential compu-

tations in the Gepner model and derive selection rules. Besides charge conservation rules

similar to those in the closed string sector, additional boundary selection rules appear, and

we illustrate these with the examples of the A1 and A2 minimal models. The selection

rules will allow us to establish that certain branes have non-trivial moduli spaces. The

exact superpotentials should be calculable given the solutions of the consistency conditions

of boundary CFT [13,14]; this is work in progress. In section 8 we summarize our results

and draw conclusions.

A point of notation: in labeling a p-brane, we always ignore its Minkowski space-

filling dimensions (for example, a D4 wraps four dimensions of the CY), but we describe

its world-volume Lagrangian in d = 4, N = 1 terms (appropriate if the brane filled all 3+1

Minkowski dimensions).

2

2. Large volume limit of the quintic

2.1. General discussion of D-branes on large volume CY

We are interested in BPS states in type II string theory described by collections of D-

branes at points on or wrapping some cycle in a Calabi-Yau manifold M . A configuration

for N coincident D-branes with worldvolume Σ wrapped on such a cycle is specified by an

embedding X : Σ →M and a U(N) gauge field A on Σ, with field strength F = dA+[A,A].The U(1) part of U(N) appears in combination with the B-field, F = F − X∗B, whereX∗B is the pullback of the NS B-field onto the worldvolume.

The conditions for supersymmetric embeddings with nonabelian fields turned on has

not been given, but they have been worked out for single D-branes in refs. [3][15], for which

the action of spacetime supersymmetry and worldvolume κ-symmetry is known [16]. A

compactification preserving supersymmetry will occur if there are constant spinors ηi on

M for each of the spacetime SUSYs. These supersymmetries transform the embedding

coordinates (and their superpartners) on the D-brane worldvolumes; they are preserved if

one can find a κ-symmetry transformation which cancels the SUSY transformation. This

condition can be written as

(1− Γ)ηi = 0 (2.1)

and those ηi which are solutions form the unbroken SUSYs. Γ is defined as follows [15].

Let Emµ be the vielbein connecting frame indices m and spacetime indices µ. We can pull

this back to the worldvolume, defining

Emα = ∂αXµEmµ (X) , (2.2)

where α is a worldvolume index for the p-brane. With this we can pull back the 10D

γ-matrices Γm:

Γα = Emα Γm . (2.3)

Define

Γ(p+1) =1

(p+ 1)!√gǫα1...αp+1Γα1...αp+1 , (2.4)

where

gαβ = ηmnEmα E

nβ (2.5)

is the induced metric on the Dp-brane. We can now write:

Γ =

√g√

g + F

∞∑

ℓ=0

1

2ℓℓ!Γα1β1...αnβnFα1β1 . . .FαnβnΓn+(p−2)/2(11) Γ(p+1) (2.6)

3

When F = 0 this can be written in the simpler, more familiar form:

Γ = ǫα1...αp+1∂α1Xµ1 . . . ∂αp+1X

µp+1Γµ1 . . .Γµp+1 (2.7)

where Γµ = Emµ Γm. The conditions in this latter case have been worked out in some detail,

as we will describe below. These conditions match those in refs. [17][18] for boundary states

of BPS D-branes in flat space with constant background fields.

Solutions to Eq. (2.1) in the presence of nonzero F have been worked out for flat,intersecting branes in refs. [17][18][15]. In the case of BPS D-branes in Calabi-Yau 3-fold

compactifications the geometric conditions implied by (2.1) (and the analog for boundary

states) have been worked out in [3][6]. These solutions fall into two classes: “A-type” branes

wrapping special Lagrangian submanifolds and “B-type” branes wrapping holomorphic

cycles. Let us describe each of these in turn.

2.1.1. B branes

“B-type” BPS branes wrap even-dimensional, holomorphic cycles in the Calabi-Yau

[3][6]. For B (even-dimensional) branes, (2.1) is solved by holomorphically embedded

curves (2-branes) and surfaces (4-branes), as well as by 0 and 6-branes with the obvious

(trivial) embeddings. We may also have gauge fields on these branes. In general the gauge

field may change the definition of a supersymmetric cycle via Eq. (2.1). However, if the

brane is wrapped around a holomorphic cycle, we can find conditions for the gauge field

to preserve the supersymmetries. In the case of N coincident D6-branes wrapping the

entire CY threefold, if we assume that the gauge fields live only in the threefold then the

SUSY-preserving gauge field must satisfy the “Hermitian Yang-Mills equations” [19]:

Fij = 0

ω2 ∧ tr F = cω3 ,(2.8)

where (i, j) and ī, ̄ are holomorphic and antiholomorphic indices, respectively, on the CY.

These equations define a “Hermitian-Einstein” connection A with curvature F . The first

equation tells us that the vector bundle is holomorphic. The second equation tells us

that the vector bundle is “ω-stable”; conversely, ω-stability guarantees a solution to these

equations [20] (c.f. chapter 4 of [21] for a discussion and definitions.)

For branes wrapped around holomorphic submanifolds of M , these equations must

be altered. The gauge fields polarized transverse to the cycle are replaced by “twisted”

4

scalars Φ which are one-forms in the normal bundle to the embedding [4], and Eq. (2.8)

becomes a generalization of the Hitchin equations for Φ and F [19].

It is believed that all topological invariants of a D-brane configuration are given by an

element of a particular K-theory group on M [22][23]. When the K-theory group and/or

the cohomology of M has torsion the K-theory interpretation is important; one may have

objects charged under the torsion. The charge can be written [22] as a generalization of

the results of [24][25]:

v(E) = ch(f!E)

√Â(M) (2.9)

Here E is a vector bundle on Σ; remember that we must extend the U(1) part of the

gauge field F by the NS B-field, so properly the vector bundle E is a polynomial in F .Let π : M → Σ be the projection onto the worldvolume and N be the normal bundleof Σ →֒ M . There is a K-theory element δ(N) which is roughly a delta function on theworldvolume and depends on N ; we can thus define f!(E) = π

∗E ⊗ δ(N). The modulispace of D-branes will not just be the moduli space of vector bundles in this K-theory class

but rather the moduli space of coherent semistable sheaves in this class [26][19]. Some

advantages of this definition through K-theory and sheaves, besides the fact that it seems

to be correct, are that it places configurations with D6-branes (gauge field configurations

onM) on an equal footing with configurations without D6-branes, and that it can describe

certain singularities which lead to sensible string compactifications.

In examples without torsion, such as the quintic, one may describe the D-brane charge

in a less esoteric fashion. Assuming the branes give rise to particles in the macroscopic

directions, for a 2n-dimensional worldvolume Σ we can write the D-brane coupling to the

RR gauge fields via the “Wess-Zumino term” as [27,24,25]:

∫

Σ

C ∧ ch(F −B)√Â(M)

Â(N)(2.10)

where

C = C(2n+1) + C(2n−1) + . . .+ C(1)

is a sum over the (k)-form RR potentials that couple to the 2n-brane.

These RR charges reduce to conventional electric and magnetic charges in the four

noncompact dimensions. Given two D-branes which reduce to particles, the most basic

observable we can study is the Dirac-Schwinger-Zwanziger symplectic inner product on

their charges,

I(a, b) = QEa ·QMb −QMa ·QEb . (2.11)

5

We will refer to this as the “intersection form” as it is closely related to the topological

intersection form for two- and four-branes. For two six-branes, from the formulas above it

is

I(a, b) =

∫ch(Fa) ch(−Fb) Â(M) . (2.12)

Finally, we quote a general theorem regarding stability (Bogomolov’s inequality [28];

c.f. [29,21]): given a variety X of dimension n and ω an ample divisor on X , then a

ω-semistable torsion free sheaf E of rank r and Chern classes ci will satisfy

∫

S

(2r c2 − (r − 1)c21

)∧ ωn−2 ≥ 0 . (2.13)

The parenthesized combination is called the “discriminant” of the sheaf and is equal to

c2(End(E)). In the special case c1(E) = 0 this amounts to requiring c2(E) ≥ 0.

2.1.2. A branes

An “A-type” BPS brane wraps a three-dimensional special Lagrangian submanifold

Σ [3]:1

ω|Σ = 0ReeiθΩ|Σ = 0 .

(2.14)

Here Ω is the holomorphic 3-form of the Calabi-Yau and θ is an arbitrary phase. Equiv-

alently to the second equation, we can require that Ω pulls back to a constant multiple

of the volume element on Σ. Furthermore the gauge field on this manifold must be flat.

A nice introduction to the general theory of these is [31]. It is shown there (and in the

references therein) that the moduli space has complex dimension b1(Σ). The space of flat

U(1) connections has real dimension b1(Σ), and ωij can be used to get an isomorphism

between T ∗Σ and NΣ; thus the deformations of Σ pair up with the Wilson lines to form

b1(Σ) complex moduli.

For three-branes, the DSZ inner product (2.11) is precisely the geometric intersection

form.

One application of these branes is the Strominger-Yau-Zaslow formulation of mirror

symmetry, a precise formulation of the idea that “mirror symmetry is T-duality” [32].

Since mirror symmetry exchanges the sets of A and B branes, an appropriately chosen

moduli space of A branes on M will be the moduli space of D0-branes on the mirror W .

1 There is some evidence that the special Lagrangian condition receives α′ corrections [30].

6

Clearly b1 = 3 for such A-branes, and SYZ argue that Σ will be a T 3 in this case. A similar

proposal was made for general B branes with bundles in [33].

Another application is the construction of N = 1 gauge theories with the help of brane

configurations. Supersymmetric three-cycles have been used to explore the strong coupling

limit by lifting the brane configurations to M-theory in [34].

Not too many explicit constructions of special Lagrangian submanifolds are known

and it appears (e.g. see [31]) that the problem is of the same order of difficulty as writing

explicit Ricci-flat metrics on a CY. A general construction we will use below is as the fixed

point set of a real involution.

2.2. General world-volume considerations

Given a system X of A or B D-branes, we can consider the system which is identical

except that it extends in the flat 3+1 dimensions transverse toM . This system will have a

d = 4, N = 1 supersymmetric gauge theory as its low-energy world-volume theory, whosedata is a gauge group GX ; a complex manifold CX parameterized by chiral superfields φ

i; a

Kähler potential K on CX ; an action by holomorphic isometries of GX on CX (linearizing

around a solution this corresponds to the usual choice of representation R of the gauge

group), and a superpotential W (a holomorphic function on CX invariant under the action

of GX). If GX contains U(1) factors, each of these can have an associated real constant

ζa (the “Fayet-Iliopoulos terms”).

In the classical (gs → 0) limit, the moduli space of this theory is the solutions of Fi =∂W/∂φi = 0 (the “F-terms”) andDa = ζa (the “D-terms”) modulo gauge transformations,

where Da is the moment map generating the associated gauge transformation (and ζa ≡ 0in the non-abelian parts of the gauge group).

We review this well-known material for a number of reasons. First, we remind the

reader that although some of our later discussion will use other realizations of this D-brane

system (for example as particles in 3 + 1 dimensions), the world-volume theories for these

other realizations are all obtained by naive dimensional reduction from the 3 + 1 theory

(if gs ∼ 0), while the 3 + 1 language makes it easy to impose supersymmetry.Second, it is known that the study of bundles and sheaves on CY three-folds is much

more complicated than that for K3; this complication has a direct physical counterpart in

the reduced constraints of N = 1 supersymmetry. The most basic example of this is thefact that – unlike the case for K3 – there is no formula for the dimension of the moduli

space of E given c(E). The main reason for this is that this dimension is not necessarily

7

constant – the moduli space can have branches of different dimension, and can depend on

the moduli of the CY as well.

Physically, this corresponds to the possibility of a fairly arbitrary superpotential in the

low energy theory. Indeed, the language of superpotentials andN = 1 effective Lagrangiansmight be the best one for these problems, much as hyperkahler geometry and hyperkahler

quotient is for instanton problems in four dimensions. Just as the self-dual Yang-Mills

equations can be regarded as an infinite-dimensional hyperkahler quotient, we might pose

the problem of rephrasing the YM equations under discussion as the problem of finding

the moduli space of an N = 1 effective theory with an infinite number of fields.The basic outlines of part of this treatment are known (see [35], ch. 6 for a very clear

discussion of the four-dimensional case). The two equations (2.8) will correspond directly

to the F-term (superpotential) constraints and the D-term constraints, respectively. In-

deed, the problem of solving Fij = 0 is a purely holomorphic problem, while it is not hard

to see that the expression F a ∧ ωn−1 is the moment map generating conventional gaugetransformations. The stability condition on the bundle is exactly the infinite-dimensional

counterpart of the usual condition in supersymmetric gauge theory for an orbit of the

complexified gauge group to contain a solution of the D-flatness conditions (e.g. see [36]).

Donaldson’s theorem proving the existence of such solutions proceeds exactly by consid-

ering the flow generated by i times the moment map to a minimum; the Uhlenbeck-Yau

generalization is quite similar (for technical reasons a different equation is used).

The other part of the story – translating the problem of finding holomorphic vector

bundles into solving constraints on a finite-dimensional configuration space, which can be

derived from a superpotential – does not seem to have been addressed in as systematic a

manner; clearly this could be useful.

In a sense the six-dimensional problem is the “universal” one which also describes the

lower-dimensional branes. Not only can their charges be reproduced, but gauge field sin-

gularities will correspond to specific lower dimensional branes (e.g. the small instanton).

Furthermore, there is a sense in which even the lower-dimensional brane world-volume

theories are six-dimensional if we include “winding strings” (by analogy to tori and orb-

ifolds, although this idea has not yet been made precise). Treating a system of N D0’s as

quantum mechanics requires neglecting these strings, which one expects to be problematic

once the separation between branes approaches the size of the space.

We now turn from these abstract ideas to our concrete example.

8

2.3. D-branes on the quintic

Perhaps the best-studied family of Calabi-Yau manifolds is the quintic hypersurfaces

in IP4. A relatively thorough discussion of these is contained in the classic paper [12]. The

moduli space of these manifolds is locally the product of b2,1 = 101 complex structure

deformations and b1,1 = 1 deformations of the complexified Kähler forms B+ iJ (where B

is the flux of the NS-NS B-field). We will be particularly interested in the Fermat quintic

P =5∑

i=1

z5i = 0 (2.15)

where zi are the homogeneous coordinates on IP4. Note that this equation has a S5 × Z45

discrete symmetry; the Z5 generators are gi : zi → ωzi and satisfy the relation∏5i=1 gi = 1,

while the S5 permutes the coordinates in the obvious way.

2.3.1. B branes on the quintic

As we have discussed, D-branes on the quintic can be described by vector bundles or

sheaves on this space. Let us denote the charge carried by a single D2p-brane wrapped

about a generator of H2p as Q2p = 1.

Transporting a D-brane configuration about closed, nontrivial cycles of the moduli

space of Kähler structures will induce an associated Sp(4, Z) monodromy on the B branes.

We will discuss the monodromy more completely in the next section, but there is already

one cycle in the moduli space which can be understood in the large volume limit: B → B+1, where B is the NS 2-form. The action on the charge Q can be seen from Eqs. (2.9),(2.10)

[37]. Mathematically this corresponds to the possibility to tensor the vector bundle V2p

with a U(1) bundle of c1 = 1. This preserves stability and the dimension of the moduli

space. Given a bundle V this operation and its inverse can be used to produce a related

bundle with −r < c1 ≤ 0: this is referred to as a “normalized” vector bundle.There is no classification of vector bundles and coherent sheaves on the quintic, but

we can write down a few examples in order to orient ourselves when discussing specific

boundary states at the Gepner point.

BPS D2-branes wrap holomorphic 2-cycles of the Calabi-Yau, the same cycles as

appear in worldsheet instanton corrections. Such cycles can have arbitrary genus and

9

arbitrary degree. Degree one rational curves are generically rigid on the quintic [38]. Non-

theless for special quintics, families may exist; for example, in the case of the Fermat quintic

(2.15), there are 50 one-parameter families essentially identical to the family [39][40]:

(z1, z2, z3, z4, z5) = (u,−u, av, bv, cv)a5 + b5 + c5 = 0; a, b, c ∈C ,

(2.16)

where (u, v) are homogeneous coordinates in IP1. Once we perturb away from the Fermat

point, these moduli are lifted and a finite number of rational curves remain [39]. This

could be described in the world-volume theory by a superpotential of the general form

W = φψ2

where φ are complex structure moduli; φ = 0 is the Fermat point; ψ are curve moduli, and

ψ = 0 a curve which exists for generic quintics.

The infinitesimal description of deformations of such cycles is as sections of the normal

bundle, which by the Calabi-Yau condition will beO(a)⊕O(b) with a+b = −2 for a rationalcurve. One might think that all one needs to find examples of families is to find examples

with a ≥ 0 or b ≥ 0, but this is not true as deformations can be obstructed. The canonicalexample is given by resolving the singularity in C4

xy = z2 − t2n . (2.17)

For n = 1 this is the conifold singularity and the “small” resolution contains a rigid IP1,

parameterized by x/(z− t) = (z + t)/y. It can be shown [41] that for n > 1 the resolutionalso contains a IP1, now with normal bundle O⊕O(−2), but the deformation is obstructedat n’th order, as could be described by the superpotential

W = ψn+1. (2.18)

Intuitively this can be seen by deforming (2.17) by a generic polynomial in t2, which splits

the singularity into n conifold singularities, each admitting a rigid IP1. If we then tune

the parameters to make these IP1’s coincide, a superpotential describing the n vacua will

degenerate to (2.18). Such singularities do appear in large families of quintic CY’s [38].2

2 (Note added in v2): The idea that the moduli space of such a curve can always be described

as the critical points W ′ = 0 of a single holomorphic function was apparently not known to

mathematicians. We thank S. Katz for a discussion on this point.

10

It turns out that the curves in (2.16) provide another example of obstructed deforma-

tions [39].3 The normal bundle of these curves is N = O(1)⊕O(−3); as dimH0(N ) = 2,there must be another obstructed deformation; call it ρ. The correct counting of curves

upon deforming away from the Fermat point can be reproduced by a superpotential ρ3.

The modulus ρ is also connected to the fact that pairs of the 50 families in (2.16) intersect

(e.g. take (2.16) and the family (av, bv, u,−u, cv) with c = 0); it describes deformationsinto the second family. All of this structure can be summarized in the superpotential

W (ρ, ψ) = ρ3ψ3 + φF (ρ, ψ) + . . . ;

where φF generalizes the φψ2 term discussed above.

Higher genus curves can generically come in families and examples can be found as

complete intersections of hypersurfaces in IP4 with the quintic. A particular example is

the intersection of two hyperplanes with the quintic [40]:

5∑

k=1

akzk =5∑

k=1

bkzk = 0 , ak, bk ∈C . (2.19)

It is easy to see that there are six independent complex parameters after rescaling the

equations. The curve is genus 6, and the area of the curve C is∫CJ = 5, where J is the

unit normalized Kähler form of IP4, i.e.

∫

IP4J ∧ J ∧ J ∧ J = 1

∫

Quintic

J ∧ J ∧ J = 5(2.20)

Thus this brane has Q2 = 5.4 There will be six additional complex moduli coming from

Wilson lines of the U(1) gauge field around the 12 cycles of the curve.

Similarly, four-branes can be obtained as the intersection with another hypersurface

in IP4. For example, the intersection of the quintic with a single hyperplane

∑

k

akzk = 0

3 (Note added in v3): We would like to thank S. Katz for explaining this example, pointing

out a mistake in our earlier draft, and suggesting the superpotential discussed here.4 See ref. [42], chapters 1 and 2 for a nice description of complete intersections in projective

spaces, and of techniques for performing the calculations we allude to here.

11

produces a four-parameter family of four-cycles S. Their volume is∫SJ = 5 and so Q4 = 5.

In addition c2(TS) = 11J2; so that the coupling of C(1) to p1/48 in eqs. (2.9),(2.10) leads

to an induced 0-brane charge of 55/24. The four-brane generically may support nontrivial

gauge field flux over two-cycles, corresponding to D2-brane charge, or instanton solutions,

corresponding to zero-brane charge. Some discussion of the moduli space of four-branes

in a Calabi-Yau can be found in [43]. By (2.13), stability of the vector bundle on the

four-brane requires Q0 > 0.

Finally we can look at the case of D6-branes wrapping the entire Calabi-Yau manifold.

In fact we will find that all of the boundary states we examine at the Gepner point will have

non-trivial six-brane charge. A single six-brane by itself will have no moduli. The U(1)

gauge field on a single 6-brane can support flux with first Chern class c1 = n corresponding

to Q4 = n. We can get the relevant bundles by restriction from U(1) bundles on IP4. The

latter have no moduli, and we will not gain any upon restriction.

We can also imagine binding D2-branes to the D6-brane, by analogy to 2 − 6 (or0 − 4) configurations in flat space. For Q6 = 1 and Q4 = 0 this appears singular; U(1)gauge fields do not support smooth instanton solutions. The brane counterpart to this is

that the 2 − 6 strings cannot be given vevs which bind the branes and give mass to therelative U(1)s. This might lead us to predict that such states, if they exist at all, exist only

as quantum-mechanical bound states. Such a state should be easily identifiable because

it appears at the junction of Coulomb and Higgs branches of the moduli space; a small

perturbation should put it on the Coulomb branch and produce two U(1) gauge fields in

the macroscopic direction. In the classical considerations of this paper, it should not show

up at all.

For Q6 > 1, we require information about vector bundles on the Calabi-Yau. A well-

known example with Q6 = 3 is deformations of the tangent bundle. This has vanishing

c1 and c2(E) = 10J giving us Q2 = 50. The dimension of the moduli space is 224. This

example can be generalized as follows. (Such generalizations are due to for example [44,45]

in the physics literature, and were previously known as “monads” in the math literature).

We consider a complex of holomorphic vector bundles

0 → A→a B →b C → 0

such that ker a = 0, im a is a subbundle of B, im b = C and define our new bundle as

E = ker b/im a.

12

For a hypersurface M in IPn, simple bundles to start with are direct sums of the line

bundles O(n) restricted to M , as in

0 → ⊕O → ⊕mi=1O(qi) → O(m∑

i=1

qi) → 0

This data allows computing the Chern classes:

cn = (

m∑

i=1

qi)n −

m∑

i=1

qni .

The dimension of the moduli space can also be computed, but this is not as easy.

A physical realization of this construction is to start with fields λi parameterizing

sections of B (e.g. the world-sheet fermions of a heterotic string theory), include a super-

potential enforcing the constraints bai λi = 0, and gauge invariances identifying λi ∼ λi+ai.

Although it is not the only place this construction appears (e.g. see [46]), the most relevant

version for present purposes is in linear (0, 2) models [45]. These constructions have the

advantage that they can be studied with conventional world-sheet techniques; a disadvan-

tage is that one requires the anomaly cancellation conditions c1 = 0 and c2(V ) = c2(T ) to

get a sensible model, so only a subset of possible V can be obtained.

The anomaly cancellation conditions also appear in D-brane constructions of the dual

type I theories as the consistency condition that the total RR charge vanishes [47]. However

in this context we need not consider branes which fill the noncompact dimensions but can

instead consider lower dimensional branes, for which these consistency conditions are not

required (a point emphasized in [5]). It seems likely that this additional freedom will lead

to a simpler theory.

Another construction of vector bundles on a CY is the Serre construction. Given a

holomorphic curve (satisfying certain conditions), this produces a rank 2 vector bundle

with a section having its zeroes on the curve. In [48] this is used to produce an example

of a vector bundle with an obstructed deformation (on a different CICY).

Finally, to conclude this section, there are a few explicit constructions of bundles on

IP4 in the literature using monads, such as the Horrocks-Mumford bundle (r = 2, c1 =

5, c2 = 10) and the bundle of Tango (r = 3, c1 = 3, c2 = 5, c3 = 5) [49], which can be

restricted to the hypersurface P = 0 to produce new examples.

13

2.3.2. A branes on the quintic

The simplest example of supersymmetric 3-cycles on the quintic are the real surfaces

Imωjzj = 0 with ω5j = 1; this was described in [3] for ω = 1. These cycles are determined

by the five phases (ω1, ω2, ω3, ω4, ω5) up to the diagonal Z5 action ωi → ωωi (which is justa remnant of the equivalence of homogeneous coordinates under complex multiplication),

so they come in a 625-dimensional irrep of the discrete symmetry S5 × Z45 .The equation

∑(ωjxj)

5 = 0, where ωixi ∈ IR, always has a unique solution for xk interms of the other real coordinates; thus the cycle is the real projective space IRP 3. The

first homotopy group is π1(IRP3) = Z2; by the discussion above (c.f. [31]) the wrapped

3-branes cannot have any continuous moduli, but they can support a discrete Z2-valued

Wilson line.

To compare these cycles with Gepner boundary states it will be useful to find their

intersection matrix. Let us choose the coordinate system z1 = 1 on IP4, so that ω1 = 1.

Regard the cycle (1, 1, 1, 1, 1) as an embedding of the coordinates x2,x3 and x4 into the

quintic with positive orientation. The other surfaces are obtained by Z45 rotation from this

one,∏5i=1 g

kii (1, 1, 1, 1, 1). Since the intersection matrix must respect the Z

45 symmetry,

it can be written as a polynomial in the generators gi and is determined by the matrix

elements

〈(1, 1, 1, 1, 1)|(1, ω2, ω3, ω4, ω5)〉 = 〈(1, 1, 1, 1, 1)|gk22 gk33 gk44 gk55 |(1, 1, 1, 1, 1)〉 (2.21)

where gkii : z → ωkiz. S5 symmetry also constrains the problem in an obvious way.There are different possibilities for intersections with the surface (1, 1, 1, 1, 1) in this

coordinate system. If ω2, ω3, ω4 and ω5 are all different from 1 there is no intersection in

this coordinate patch. If only three of them are different from 1 there is exactly one inter-

section in this coordinate patch and the intersection has the signature sgn Imω2Imω3Imω4

assuming that ω5 = 1. If the two surfaces intersect on a higher dimensional locus the in-

tersection number has to be calculated by a small deformation of one of the two surfaces.

This deformation has to be normal to both surfaces. Because of the special Lagrangian

property of the undeformed surfaces this “normal bundle” of the intersection locus can be

identified with its tangent space. The intersection number is then given by the number of

zeros of a section of the tangent bundle of the intersection locus.

For example, in the case that exactly two ωj ’s are not 1 the intersection locus is a circle.

A circle can have a nowhere vanishing section of its tangent bundle and the intersection

14

number in this coordinate patch is 0. As another example, let precisely one ωj 6= 1. Theintersection locus is then an IRP 2. A section of its tangent bundle has one zero, as can be

seen by modding out the ’hedgehog configuration’ of an S2 by Z2. The orientation of this

intersection is given by the intersection in the remaining complex dimension, i.e. by Imωj .

In order to compute the full intersection we must look at all possible patches.

This can be done by using the constraint∏5i=1 gi = 1 to rewrite (1, ω2, ω3, ω4, ω5) as

(ω−12 , 1, ω−12 ω3, ω

−12 ω4, ω

−12 ω5) and so on. We then add all of the intersection numbers

for all of these patches. Thus, although we find that 〈(1, 1, 1, 1, 1)|(1, ω, ω, ω, ω)〉 = 0 inthe z1 = 1 coordinate patch, the total intersection number – the coefficient of

∏5i=2 gi in

the intersection matrix – is 1. Another example is the intersection of (1, 1, 1, 1, 1) with

(1, ω, ω, ω, 1) which gives a circle in the patch z2 = 1 and a point in the patch z1 = 1.

A simple general formula that matches all of these results is

IIRP 3 =5∏

i=1

(gi + g2i − g3i − g4i ). (2.22)

3. Stringy geometry

Type IIb string compactification on a general CY threefold M leads to an N = 2,d = 4 supergravity with b2,1 + 1 vector fields (b2,1 vector multiplets plus the graviphoton)

and b1,1 + 1 hypermultiplets (including the 4d dilaton); in IIa these identifications are

reversed. The most basic physical observables which reflect the structure of M are those

described by the special geometry of the vector multiplets. This geometry is determined

by a prepotential FK of Kähler deformations in the IIa case, and by the prepotential Fc

for complex structure deformations in the IIb case.

A fundamental result from the study of the worldsheet sigma model is that Fc can be

determined entirely from classical target space geometry; it receives no worldsheet quantum

(α′) corrections. Let us then discuss the complex structure moduli space. Choose a basis

for the 3-cycles Σi ∈ H3(M,ZZ) (where i = 0, . . . , b2,1, b2,1 + 1, . . . , 2b2,1 + 2), so that theintersection form ηij = Σi ·Σj takes the canonical form ηi,j = δj,i+b2,1+1 for i = 0, . . . , b2,1(an a cycle with a b cycle). The b2,1+1 vector fields come from reducing the RR potential

C(4) on the a cycles, while the b cycles produce their d = 4 electromagnetic duals. Thus a

three-brane wrapped about the cycle Σ =∑

iQiΣi has (electric,magnetic) charge vector

Qi. Note that H3(X) forms a nontrivial vector bundle over the moduli space Mc of

15

complex structures; a given basis in H3(X,ZZ) will have monodromy in Sp(b3,ZZ) as it is

transported around singularities in Mc.The primary observables are the periods of the holomorphic three-form,

Πi =

∫

ΣiΩ.

In N = 2 language these are the vevs of the scalar fields in the corresponding vectormultiplets. The a-cycle Πi’s can be used as projective coordinates on the moduli space;

the b-cycle periods then satisfy the relations Πj = ηij∂F/∂Πi. If we fix (for example)Π0 = 1 to pass to inhomogeneous coordinates, the related vector field is the graviphoton.

These periods determine the central charge of a three-brane wrapped about the cycle

Σ =∑iQi[Σ

i]:

Z =

∫

Σ

Ω = QiΠi.

Thus the mass of a BPS three-brane is [50]:

mQ = c|Z| = c|Q ·Π| (3.1)

where c is independent of Q. If we use four-dimensional Einstein units for m, it is c =

1/gs(∫Ω ∧ Ω̄)1/2.

In contrast to Fc, FK receives world-sheet instanton corrections to the classical com-

putation. The exact worldsheet result can be obtained by mirror symmetry: FK for IIa on

M is equal to Fc for IIb on the mirror W to M . Of course this requires a map between the

periods of M and W . This analysis has been carried out for the quintic in [12] (see [51]

for a summary) and we will quote the result in this case.

The mirror W to the quintic threefold M can be obtained [52] as a Z35 quotient of a

special quintic

0 =

5∑

i=1

z5i − 5ψz1z2z3z4z5 .

The transformation ψ → αψ with α5 = 1 can be undone by the coordinate transformationz1 → α−1z1 and thus the complex moduli space of W ’s can be parameterized by ψ5. Thisis an “algebraic” coordinate, which although not directly observable, does appear naturally

in the world-sheet formulations [53,54].

16

The moduli space M has three singularities, about which the three-cycles in W willundergo monodromy. Each singularity has physical significance. First, ψ5 → ∞ is the“large complex structure limit” mirror to the large volume limit. In this limit [51]

(5ψ)−5 → e2πi(B+iJ) , (3.2)

where B is the NS B-field flux around the 2-cycle forming a basis of H2(M), and J is

the size of that 2-cycle. Next, ψ5 → 1 is a conifold singularity; here a wrapped three-brane becomes massless [55]. This turns out to be mirror to the “pure” six-brane [56,57].

Finally, at ψ5 = 0 the model obtains an additional Z5 global symmetry; this is an orbifold

singularity of moduli space. The Gepner model (3)5 lives at this point in Kähler moduli

space of M [53].

Each singularity in M gives a noncontractible loop, which is associated with a mon-odromy on the basis of 3-cycles in W (or even homology in M) and thus on the periods.

We let A be the monodromy induced by ψ → αψ around ψ = 0; clearly A5 = 1. Twill be the monodromy induced by going once around the conifold point, and B will be

the monodromy induced by taking ψ → α−1ψ around infinity. These satisfy the relationB = AT . One may make the physics associated with a given singularity manifest by

choosing variables (the periods) for which the associated monodromy is simple.

In our case the periods Πi satisfy a Picard-Fuchs differential equation of hyperge-

ometric type. Since b3 = 4 it is fourth order and quite tractable. There will be four

independent solutions and as per the discussion above, we generally want to choose a basis

making one of the monodromies simple. Two such bases are particularly natural. The first

is the large volume basis which we will denote (Π6,Π4,Π2,Π0)t. Up to an upper triangular

transformation this is determined by the asymptotics as ψ5 → ∞

Π6Π4Π2Π0

→

−56 (B + iJ)3−52 (B + iJ)2

B + iJ1

. (3.3)

The coefficients correspond to the classical volumes of the cycles. The signs were chosen

so that the supersymmetric brane configurations have positive relative charges. We will

derive the monodromy below.

The other natural basis for us makes the monodromy A simple, and is appropriate for

describing the Gepner point. If we choose a solution ΠG0 (ψ) analytic near ψ = 0, the set

of solutions

ΠGi (ψ) = ΠG0 (α

iψ) (3.4)

17

will provide a basis with the single linear relation 0 =∑4i=0 Π

Gi . It turns out that the

0-brane period Π0 (the solution ω̃0 of [12], equation (3.15)) is analytic near ψ = 0 and

thus we can set ΠG0 = Π0 and define the others using (3.4). We then (as in [12]) choose

the period vector (ΠG2 ,ΠG1 ,Π

G0 ,Π

G4 )t. In this basis, the three monodromy matrices are 5

AG =

−1 −1 −1 −11 0 0 00 1 0 00 0 1 0

TG =

1 4 −4 00 0 1 00 −1 2 00 4 −4 1

BG =

−1 −7 5 −11 4 −4 00 0 1 00 −1 2 0

(3.5)

In [12], the relation between the large volume and Gepner bases proceeds through

a third basis which we will call Π3, which is naturally described by a particular basis of

3-cycles in W . The intersection form in this basis has the canonical form η13 = η24 = −1,and the T monodromy is simple: Π3i → Π3i + δi,2Π34. Thus Π34 is the vanishing cycle at theconifold and Π32 is its dual. This turns out to be enough information to relate it to the

Gepner basis uniquely up to a remaining SL(2, Z) acting on Π31 and Π33, which we may

fix arbitrarily. One then finds a transformation of Π3 to a basis satisfying (3.3). This is

an SL(2, Z) transformation of the type which was unfixed in the previous step; so the Π3

basis has no significance intrinsic to our problem of relating ΠG to the large-volume basis.

Thus we will merely quote the final result for this change of basis, which is:

Π =MΠG Q = QGM−1 A =MAGM−1 . . .

M = L

0 −1 1 0−3

5−1

5215

85

15

25 −25 −15

0 0 1 0

(3.6)

Here Q and QG are the charge vectors in the large-radius and Gepner basis respectively.

(In the notation of [12],M = KNm: with K a matrix taking the vector (Q4,−Q6, Q2, Q0)of their conventions to our conventions; and N taken with a′ = b′ = c′ = 0.) The matrix

5 There is a typo in table I in [12] as published in Nuclear Physics B.

18

L is an as-yet undetermined Sp(4, Z) ambiguity in the Q2 and Q0 charges of the six- and

four-branes:

L =

1 0 −b −c0 1 a b0 0 1 00 0 0 1

with (a, b, c) integers (the (a′, b′, c′) of [12]).

Given the classical intersection form η in the large-radius limit, we can now determine

the intersection form in the Gepner basis:

ηg =M−1η(M−1)t =

0 −1 3 −31 0 −1 3−3 1 0 −13 −3 1 0

, (3.7)

where η14 = −η41 = −η23 = η32 = 1 from [12].6 L does not enter since it is symplectic,and so preserves η. ηg has determinant 25 and thus the Gepner basis is not canonically

normalized; this point will not be important for us.

We want to better understand the ambiguity L. We can start by comparing the

monodromy B with our expectations from the large volume limit. One may define a basis

of charges such that ΓRRk is the charge under the RR potential C(k+1), with the switch

in four- and six-brane charge as in (3.3). In this basis the effect of the shift B → B + 1follows from Eq. (2.10):

BL =

1 1 −52

−56

0 1 −5 −520 0 1 10 0 0 1

. (3.8)

The factors 1/2 and 1/6 in this expression come from expanding the exponential (they can

also be seen in (3.3)) and indicate that in this basis the charges are not integers.

The B monodromy in the Π basis (3.3) is

B =

1 1 3− a −5− 2b0 1 −5 −8 + a0 0 1 10 0 0 1

. (3.9)

Eqs. (3.8) and (3.9) agree if a = 11/2 and b = −25/12, i.e. if we make a non-integralredefinition of the charge lattice. The explanation of this is that the intersection form in

6 The signs Σ6 · Σ0 = +1 and Σ4 · Σ2 = −1 in the large volume intersection form η follow

from the definition (2.12).

19

the conventions leading to (3.8) is actually not canonical, because it includes the other

terms in (2.10). If we act on the basis (3.3) with the matrix L(a = 11/2, b = −25/12, c),we can see that the charges are modified in precisely this way. The modification due to b

comes from the  term in (2.12)(as c2 = 50 for the quintic). a induces a two-brane charge

on the four-brane and might come from c1 of its normal bundle. These effects were referred

to in [19] as the “geometric Witten effect”.

The most interesting ambiguity comes from c which induces zero-brane charge on the

six-brane. In [12] this was attributed to the sigma model four-loop R4 correction in the

bulk Lagrangian. In the D-brane context, one possibility is that this comes from an as yet

unknown term at this order in the D-brane world-volume Lagrangians. We should also

keep in mind that the intersection form we are computing involves the bulk propagation

of the RR fields between the branes, so another possibility is that it comes from a partner

to the R4 term in the bulk Lagrangian which affects the RR kinetic term in a curved

background.

In [12], the redefinition L was used to make the charge basis integral, but an overall

Sp(4, Z) ambiguity was left over. It is in general more useful to have an integer charge

basis so we will follow this procedure (this was already done implicitly as we took integer

coefficients in the change of basis). We can resolve most of the Sp(4, Z) ambiguity by

calling the state which becomes massless at the mirror of the conifold point a “pure” six-

brane with large volume charges (1 0 0 0), following [56,57]. This determines b = c = 0. A

geometrical argument for this is that any fluxes on the six-brane would produce additional

contributions to its energy. If there is a line from the large volume limit to the conifold

point along which the six-brane becomes massless with no marginal stability issues, this

argument will presumably be valid. Another argument is that we will find this state as

a Gepner model boundary state with no moduli, as is appropriate for a pure six-brane.

Finally, this choice simplifies the charge assignments for the other boundary states.

We still have the ambiguity in a to fix. As it happens this does not enter into the

results we discuss, so we have no principled way to do this. We will simply set it to zero.

4. Boundary states in CFT

4.1. Some results from boundary conformal field theory

A CFT on a Riemann surface with boundary requires specifying boundary conditions

on the operators. For sigma models these conditions can be derived by imposing Dirichlet

20

and/or Neumann boundary conditions directly on the sigma model fields. For more general

CFTs we do not have a nice Lagrangian description; so the construction, classification,

and interpretation of boundary conditions is not as straightforward. (See [58,7,59,8] and

references there for recent work in this direction.)

If the CFT has a chiral symmetry algebra one may simplify the problem by demanding

that the boundary conditions are invariant under the symmetry. We can start with the Vi-

rasoro algebra which must be preserved (particularly in string theory where the symmetry

is gauged). Let the boundary be at z = z̄ in some local coordinates. Reparameterizations

should leave the boundary fixed, so we must impose T = T̄ . If the remaining symmetry

algebra is generated by chiral currents W (r) with spin sr, then the boundary conditions

are

W (r) = ΩW̄ (r)Ω† , (4.1)

where Ω is an automorphism of the symmetry algebra.

We are interested in describing BPS D-branes which preserve N = 1 spacetime SUSY.The closed-string sector will have at least N = (2, 2) worldsheet SUSY and the boundaryconditions must preserve a diagonal N = 2 part [60,61]. Eq. (4.1) leads to two classes ofboundary conditions [6]: the “A-type” boundary conditions

T = T̄ , J = −J̄ , G+ = ±Ḡ− , (4.2)

and the “B-type” boundary conditions

T = T̄ , J = J̄ , G+ = ±Ḡ+ . (4.3)

These conventions correspond to the open-string channel where the boundary propagates

in worldsheet time. For Calabi-Yau compactification at large volume, A-type boundary

conditions correspond to D-branes wrapped around middle-dimensional supersymmetric

cycles; and B-type boundary conditions to D-branes wrapped around even-dimensional

supersymmetric cycles [6].

A CFT on an annulus can also be studied in the closed-string channel where time flows

from the one boundary to the other. The boundaries appear as initial and final conditions

on the path integral and are described in the operator formalism by “coherent” boundary

states [62,63]. The boundary conditions (4.1) can be rewritten in the closed-string channel

as operator conditions on these boundary states; for example

Jn = J̄−n A type

Jn = −J̄−n B type.

21

The relative sign change from (4.2),(4.3) can be understood as the result of a π/2 rotation

on the components of the spin one current; it means that the A-type states are charged

under (c, c) operators and the B-type under (c, a) operators.

The solution to these conditions [64,65] are linear combinations of the “Ishibashi

states”:

|i〉〉Ω =∑

N

|i, N〉 ⊗ UΩ|i, N〉 . (4.4)

Here |i〉 is a highest weight state of the extended chiral algebra; the sum is over all de-scendants of |i〉; and U is an anti-unitary map with U |i, 0〉 = |i, 0〉⋆ and UW̄ (r)n U † =(−1)srW̄ (r)n .

Modular invariance requires that calculations in either channel have the same result.

This gives powerful restrictions on possible boundary states. In particular one requires

that a transition amplitude between different boundary states can be written as a sensible

open-string partition function, via a modular transformation. For rational CFTs with

certain restrictions, Cardy [13] showed that the allowed linear combinations of Ishibashi

states (4.4) are:

|I〉〉Ω =∑

j

BjI |j〉〉Ω =∑

j

SjI√Sj0

|j〉〉Ω . (4.5)

If χj is a character of the extended chiral algebra, then Sji is the matrix representation

of the modular transformation τ → −1/τ . In this notation capital and lower-case lettersdenote the same representation; we use capital letters to denote this particular linear

combination of Ishibashi states. We may also associate a bra state to the representation

I∨ conjugate to I:

Ω 〈〈I∨| =∑

j

Ω 〈〈j|BjI . (4.6)

These boundary states are in one-to-one correspondence with open-string boundary con-

ditions which we will label the same way. Cardy argued that the open-string partition

function was determined by the fusion rule coefficients. Let worldsheet time and space

be labeled by τ and σ respectively; and let the boundary run from σ = 0 to σ = π, and

the boundary conditions be I∨ and J , respectively. Then the number of times that the

representation k appears in the open-string spectrum is precisely the fusion rule coefficient

NkIJ ; in other words, the open-string partition function will be

ZI∨J =∑

k

NkIJχk . (4.7)

22

4.2. The Gepner model in the bulk

Gepner models [66,67] (see also [68] for a quick review) are exactly solvable CFTs which

correspond to Calabi-Yau compactifications at small radius [53]. They are tensor products

of r N = 2 minimal models together with an orbifold-like projection that couples thespin structures and allows only odd-integer U(1) charge. We will review their construction

here. For simplicity we will discuss theories with d + r = even, where d is the number of

complex, transverse, external dimensions in light cone gauge.

Our building blocks are the N = 2 minimal models at level k; these are SCFTs withcentral charge c = 3k

k+2< 3 [69,70,71,72]. The superconformal primaries are labelled by 3

integers, (l,m, s) with

0 ≤ l ≤ k; |m− s| ≤ l; s ∈ {−1, 0, 1}; l +m+ s = 0 mod 2 . (4.8)

The integers l and m are familiar from the SU(2)k WZW model and can be understood

from the parafermionic construction of the minimal models [73,74]. s determines the spin

structure: s = 0 in the NS sector; and s = ±1 are the two chiralities in the R sector.7 Theconformal weights and U(1) charges of these primary fields are:

hlm,s =l(l + 2)−m2

4(k + 2)+s2

8,

qlm,s =m

k + 2− s

2.

(4.9)

The N = 2 chiral primaries are clearly (l,±l, 0) in the NS sector. The related Ramondsector states (l,±l,±1) can be reached by spectral flow. The minimal models can also bedescribed by a Landau-Ginzburg model of a single superfield with superpotential Xk+2

[75,76,77,78,79]. At the conformal point X l = (l, l, 0) and the Landau-Ginzburg fields

provide a simple representation of the chiral ring.

The N = 2 characters and their modular properties are described in [80,81,66,67]; wewill follow the notation in [66,67]. One extends the s variable to take values in Z4. The NS

characters are labelled by s = 0, 2 and the different values of s denote opposite Z2 fermion

number. The contribution from the NS primary is in χl,m,0. Similarly, in R sector s = ±1denotes contributions from opposite fermion number: the s = 1(s = 3) character includes

the contribution from the s = 1(s = −1) Ramond-sector primary. These characters are

7 The variable m in [74], in sec. 2.1 of [66], and sec. 4 of [67], is what we are calling m− s.

23

actually defined in the range l ∈ {0, · · · , k}, m ∈ Z2k+4 and s ∈ Z4, where l+m+s = even.They obey the identification χlm,s = χ

k−lm+k+2,s+2 by which the fields can be brought into

the range (4.8).

Not every c = 9 tensor product of minimal models will give a consistent string com-

pactification with 4d spacetime SUSY. We must find a reasonable GSO projection, and we

must project onto states with odd integer U(1) charges [60]. We must then add “twisted”

sectors in order to maintain modular invariance. The resulting spectrum is most easily

represented by the partition function, for which we require some notation. We will tensor

r minimal models at level kj with the CFT of flat spacetime. The latter also has a N = 2worldsheet SUSY in our case, and we denote the characters by the indices i. The vector

λ = (l1, · · · , lr) gives the lj quantum numbers and the vector µ = (m1, · · · , mr; s1, · · · , sr),the charges and spin structures. Now define βj=1,...,r to be the charge vector with a two at

the position of sj, and all other entries zero; and define β0 to be the charge vector with all

entries one. The modular invariant partition function in light cone gauge can be written

as [66,67]:

Z =∑

(i,̄i),λ,µ

∑

b0,bj

δβ(−1)b0χi,λ,µ(q)χī,λ,µ+b0β0+∑jbjβj

(q̄) , (4.10)

Here χi,λ,µ is the character for the r minimal models specificed by λ, µ and for the character

of the flat transverse spacetime coordinates (labelled by i). In the sum, b0 = 0, · · · , 2K−1,bj = 0, 1 and K = lcm{2, kj + 2}. δβ is a Kronecker delta function enforcing both oddintegral U(1) charge and the condition that all factors of the tensor product have the same

spin structure.

The kth minimal model has a Zk+2 × Z2 symmetry [66,82] which acts as:

gφlm,s = e2πi m

k+2φlm,s,

hφlm,s = (−1)sφlm,s .(4.11)

With the above projection, all Z2 symmetries have the same action on a given state and are

identified. The remaining Z2 symmetry acts only on R states by reversing their sign. The

Zk+2 symmetry is correlated with the U(1) charge. In particular, the diagonal generator

G =∏j gj is the identity for integral U(1) charges. The Gepner model is an orbifold theory;

the orbifold group H is the group generated by G. The remaining discrete symmetry is

⊗ri=1Zkr+2/H. For example, the (k = 3)5 model is an orbifold by the diagonal Z5 of(Z5)

⊗5 .

24

4.3. Boundary states in the Gepner model

It is difficult to construct the most general boundary state for the Gepner model,

because the Gepner model is not rational. Following [7], we will consider states which

respect the N = 2 world-sheet algebras of each minimal model factor of the Gepner modelseparately, and can be found by Cardy’s techniques. These might be called “rational

boundary states.” They are labeled according to Cardy’s notation by α = (Lj ,Mj, Sj)

and an automorphism Ω of the chiral symmetry algebra. In our case there are two choices

of Ω giving either A- or B-type boundary conditions; Ω must have the same action on

every factor of the tensor product.

Recknagel and Schomerus [7] proved the modular invariance of A- and B-type bound-

ary states with internal part:

|α〉〉 = 1κΩα

∑

λ,µ

δβδΩBλ,µα |λ, µ〉〉Ω . (4.12)

The coefficients are:

Bλ,µα =

r∏

j=1

1√√2(kj + 2)

sin(lj, Lj)kj√sin(lj, 0)kj

eiπ

mjMj

kj+2 e−iπsjSj

2 , (4.13)

a result of eq. (4.5) for the minimal models and the extra coefficient κΩα described in the

appendix. Here

(l, l′)k = π(l + 1)(l′ + 1)

k + 2.

δΩ denotes the constraint that the Ishibashi state |λ, µ〉〉Ω must appear in the closed stringpartition function (4.10). For A-type boundary states this is no constraint as the Ishibashi

states are already built on diagonal primary states and δβ already enforces that total U(1)

charge is integral. However, the B-type Ishibashi states have opposite U(1) charge in the

holomorphic and antiholomorphic sector, and these only appear as a consequence of the

GSO projection; so the δB constraint requires that all the mj are the same modulo kj +2.

Finally, an integer normalization constant C has to be included in κΩα to get the correct

normalization for the open-string partition function.

It is easy to see from eqs. (4.12),(4.13) that the action of the Zkj+2 (Z2) symmetries

is Mj → Mj + 2 (Sj → Sj + 2). As a result of the δβ constraint, the two physicallyinequivalent choices for Sj are S =

∑Sj = 0, 2 mod 4. The Sj = odd case seems to be

inconsistent because their RR-charges do not fit into a charge lattice together with the

25

S = even states; thus they will violate the charge quantization conditions8. In the end,

due to the Z2 symmetry, it is enough to consider only boundary states with S = 0. A

boundary state can be written as

gM12

1 · · · gMr2r h

S2 |L1 · · ·Lr〉Ω := |L1 · · ·Lr;M1 · · ·Mr;S〉Ω =

gM1−L1

2

1 · · · gMr−Lr

2r h

S2 |L1 · · ·Lr;M ′1 = L1 · · ·M ′r = Lr;S′ = 0〉Ω .

For B-type boundary states, the δβ constraint in eq. (4.12) implies in addition that the

physically inequivalent choices of Mj can be described by the quantity

M =∑

j

K ′Mjkj + 2

,

where K ′ = lcm{kj + 2}.We will be interested in counting the number of moduli for a D-brane state; these

will be the massless bosonic (i.e. NS) open-string states. To find their contribution to

the open-string partition function, it is enough to examine the NS-NS part of a transition

amplitude in the internal dimensions. The reason is that the (open-string) NS characters

arising from the modular transformations of the RR part of the transition amplitude come

with an insertion of (−1)F [80,81]. With this in mind, a calculation similar to that in [7]leads to9

ZAαα̃(q) =1

C

NS∑

λ′,µ′

K−1∑

ν0=0

r∏

j=1

Nl′j

Lj ,L̃jδ(2kj+4)

2ν0+Mj−M̃j+m′jχλ

′

µ′(q) , (4.14)

and

ZBαα̃(q) =1

C

NS∑

λ′,µ′

δ(K′)M−M̃

2+∑

K′2kj+4

m′j

r∏

j=1

Nl′j

Lj ,L̃jχλ

′

µ′(q) . (4.15)

(Here δ(n)x is one when x = 0 mod n and zero otherwise.) This shows that only a U(1)

projection and the SU(2)k fusion rule coefficients constrain the open string spectrum of

B-type boundary states; these states are much richer as a consequence.

8 The amplitude between a S = odd boundary state and a S̃ = even boundary state also has

interchanged roles of R- and NS-states in the open string sector.9 N l

L,L̃are the SU(2)k fusion rule coefficients [83]: they are one if |L − L̃| ≤ l ≤ min{L +

L̃, 2k − L− L̃} and l + L + L̃ = even, and zero otherwise; note that our indices thus differ from

those in [83] by a factor of two.

26

The condition that two D-brane boundary states |α〉〉 and |α̃〉〉, with the same externalpart, preserve the same supersymmetries is [7]:

Q(α− α̃) := −S − S̃2

+r∑

j=1

Mj − M̃jkj + 2

= even . (4.16)

To explore the charge lattice of the boundary states, and to find the geometric inter-

pretation of given boundary states, we wish to calculate the intersection (2.11)(2.12) of

our branes. The CFT quantity which computes this is IΩ = tr R(−1)F in the open stringsector [11]. The best way to do this is to start in the closed string sector and to do a

modular transformation to the open string sector. In the closed string sector this trace

corresponds to the amplitude between the RR parts of the boundary states with a (−1)FLinserted. The calculation is done in the Appendix and the result for A-type boundary

states is:

IA =1

C(−1)S−S̃2

K−1∑

ν0=0

r∏

j=1

N2ν0+Mj−M̃jLj ,L̃j

. (4.17)

For B-type boundary states,

IB =1

C(−1)S−S̃2

∑

m′j

δ(K′)M−M̃

2+∑

K′2kj+4

(m′j+1)

r∏

j=1

Nm′j−1Lj ,L̃j

. (4.18)

The intersection matrix depends only on the differences M − M̃ as was required by thediscrete symmetry. We also see that the Z2 action S → S +2 changes the orientation of abrane.

In the next section we will rewrite these formulas in a more compact notation and use

them to identify the charges of the boundary states.

4.4. D-branes on K3 and the Mukai formula

For compactifications with N = 4 worldsheet supersymmetry, the index in the Ra-mond sector is directly related to the number of marginal operators in the NS sector. We

now use this to give a CFT proof of Mukai’s formula [84,19] for the dimension of the moduli

space of 1/2-BPS D-brane states.

K3 compactifications are geometric throughout their moduli space [85]. The BPS

D-brane states in these compactifications are described by coherent semistable sheaves E

27

[19] which can be labelled by the Mukai vector [84,19]. In terms of the rank r and Chern

classes ci of E, this is

v(E) =

(r, c1,

1

2c21 − c2 + r

)

∈ H0(M,ZZ)⊕H2(X,ZZ)⊕H4(M,ZZ)(4.19)

There is a natural inner product on the space of Mukai vectors:

〈(r, s, ℓ), (r′, s′, ℓ′)〉 = s · s′ − rℓ′ − ℓr′ (4.20)

where s · s′ is defined by the natural intersection pairing of 2-cycles on M . In fact this isjust (minus) the intersection form (2.12).

Mukai’s theorem [84] states that the complex dimension of the moduli space of an

irreducible coherent sheaf E is:

dimension = 〈v(E), v(E)〉+ 2. (4.21)

We now argue that this follows from the relation

tr a,a(−1)F = 〈v(Ea), v(Ea)〉 (4.22)

and general properties of supersymmetry. First, only two d = 2, N = 4 representationshave nonvanishing Witten indices [86,87]. We list them below together with the NS weights

related by spectral flow:

identity rep. : (h = 0, ℓ = 0)NS −→ (h = 1/4, ℓ = 1/2)R tr(−1)F = −2“massless′′ rep. : (h = 1/2, ℓ = 1/2)NS −→ (h = 1/4, ℓ = 0)R tr(−1)F = 1 ,

(4.23)

where ℓ is the SU(2)R isospin. The identity representations lead to world-volume d = 6,

N = 1 (or d = 4, N = 2) gauge multiplets, while the massless representations lead toworld-volume half-hypermultiplets, so there will be one complex scalar in the open-string

sector for each massless multiplet.

Let there be Ng identity and Nm massless multiplets; then the Witten index is

tr (−1)F = Nm − 2Ng. (4.24)

Using (4.22) we find that (4.21) will be true if the world-volume theory has a (Higgs

branch) moduli space of complex dimension Nm−2Ng+2. This moduli space is essentially

28

determined by the d = 6, N = 1 world-volume supersymmetry: it is the hyperkählerquotient of the configuration space by the subgroup G of the gauge group which acts non-

trivially on the hypermultiplets. The resulting space has complex dimension Nm−2dimG.Now, any brane configuration will have an overall U(1) acting trivially whose partners

in the vector multiplet are the center of mass position of the brane; if more U(1)s act

trivially we will have more center of mass moduli, so such a configuration must correspond

to a reducible bundle. Therefore dimG = Ng − 1 for an irreducible bundle and we haveproven (4.21).

4.5. Generalizations

Mukai’s theorem used the Hirzebruch-Riemann-Roch formula together with special

properties of K3 surfaces; these properties allowed one to extract the dimension of the

moduli space of a bundle directly from the holomorphic Euler characteristic. We have a

similar statement for CY threefolds if we keep track of both chiralities separately. The self-

intersection number of a brane on a threefold is of course zero, but we can get non-trivial

statements if we consider the intersection of two different branes.

For example, consider the index of the Dirac operator on the bundle E. Since the

world-volume is Kähler this is

ind /D =3∑

i=0

(−1)idimHi(M,E) = χ(E)

which is the holomorphic Euler characteristic. By the Hirzebruch-Riemann-Roch formula,

χ(E) =

∫

M

ch(E)Td(TM) . (4.25)

Here

ch(E) = r + c1(E) +1

2

(c21(E)− 2c2(E)

)+

1

6

(c31(E)− 3c1(E)c2(E) + 3c3(E)

)+ . . . ,

and

Td(TM) = 1 +c2(TM)

12+ . . . = 1− p1(TM)

24+ + . . .

Thus on a threefold, Td(M) = Â(TM), and combining eqs. (4.25) and (2.12), we find:

ind /D = 〈D6, D(E)〉 = tr D6,D(E)(−1)F , (4.26)

29

where D(E) is the D-brane representation or generalized Mukai vector for E.

On the other hand, the Ramond ground states which contribute to the open string

index are exactly the fermion zero modes which contribute to the index of /D. In the type

I case where E is a gauge bundle with vevs entirely in an SU(3) subgroup and with the

gauge connection equal to the spin connection, c1(E) = 0; this gives a brane picture of the

standard result

Ng = (# of generations) =

∫

M

c32

for this case. If we are interested not in the bulk gauge theory on 9-branes in type I but

in a gauge theory on a brane B intersecting another brane A, the generalization is that

the number of generations (with respect to the B gauge group) associated with the brane

A is the intersection form 〈A,B〉. For B-type branes this follows from eq. (2.12) and theHirzebruch-Riemann-Roch theorem for the bundle E(A)∗ ⊗E(B); for A-type branes eachintersection contributes a chiral multiplet with chirality given by the sign of the intersection

[17].

5. Discussion of the 35 model

Let us apply these results to the example to model (k = 3)5, the Gepner point in the

moduli space of the quintic. We will consider boundary states labelled by Lj ∈ {0, 1},0 ≤ Mj < (2k + 4) = 10, and S = 0. Let the Z45 symmetry be generated by theoperators gj taking Mj →Mj +2, and satisfying g1 · · · g5 = 1. Note that g1/2j which takesMj →Mj + 1 is well-defined for these states (using the identifications on LMS, it relatesbranes to antibranes).

We will be particularly interested in computing the intersection forms (4.17) and

(4.18), as we will be able to use them to extract the charges and open string spectrum

for a given brane. The main advantage of considering these quantities over the charges

themselves is that they are canonically normalized, as already noted in [1].

We can consider the intersection form as a matrix I acting on the space of boundary

states; since it commutes with Z45 it can be written as a function of the generators gi. The

main content of formulae (4.17) and (4.18) is contained in the SU(2) fusion rule coefficients.

30

In these equations the labels Mj , M̃j can be thought of as indices of a matrix acting on

the states. The particular fusion coefficients we will need are:10

NMj−M̃j00 → (1− g4j ),

NMj−M̃j01 → g

12

j (1− g3j ) = N00 g12

j (1 + g4j );

NMj−M̃j11 → (1 + gj − g3j − g4j ) = N01 g

12

j (1 + g4j ).

(5.1)

These various fusion matrices are related by successive multiplication with g12

j (1 + g4j ), so

we can express the RR charges of all our boundary states in terms of those for Q(|00000〉Ω).By eq. (4.16) there are two cases of pairs of branes preserving a common susy. If

the total ∆L is even (so integral powers of g appear), a pair with ∆M = ∆S = 0 (brane

and brane) will preserve susy. If the total ∆L is odd (powers g5k+5/2 appear), a pair with

∆M = 5 and ∆S = 2 (brane and anti-brane) will preserve susy.

In the case that the two D-branes are both A-type or B-type, the massless open string

spectrum can also be expressed in terms of the fusion coefficients. It is easy to see from

(4.14) and (4.15) that if the two boundary states are the same, there is exactly one vacuum

and one spectral flow operator in the open string channel; if they are not the same, neither

state propagates. This means that the unbroken worldvolume gauge group is (the center-

of-mass) U(1), and the brane can be viewed as a single object (a priori, it still might be a

bound state).

The SUSY-preserving moduli of the D-branes are constructed from chiral vertex op-

erators. The Witten index counts these operators albeit with a sign depending on their

chirality. In our explicit CFT calculation we can remove this sign by hand, and thus the

total number of chiral fields can be calculated using (4.17) and (4.18) with the fusion ma-

trices replaced by their absolute values.11 We can again write this “modified” matrix as a

polynomial PΩ(gj) in the shift matrices gj . For example, the matrix for boundary states

|11111〉B is:PB(g) = (1 + g + g

3 + g4)5 . (5.2)

If spacetime supersymmetry is preserved, the chiral fields have integer U(1) charges, and

are related to antichiral fields by spectral flow. In particular charge-2 chiral fields in ZΩαα̃,

are related to charge-−1 antichiral fields in ZΩαα̃; the latter are the hermitian conjugate of

10 The coefficients for m > l are defined in the Appendix.11 In other words, we define Nm

LL̃= +N−m−2

LL̃, rather than the opposite sign in the Appendix.

31

charge-1 chiral fields in ZΩα̃α. Thus∑kmk in the open-string channel will be a multiple of

5 for marginal, chiral vertex operators. Examination of the fusion coefficients in (4.17) and

(4.18) reveals that the number of massless chiral superfields is given by counting terms in

1

2(PΩ(gj)− 2)

with the total power of g being a multiple of 52 .

Applying these statements to eq. (5.2) shows that the D-brane described by |11111〉Bhas 101 marginal operators. This particular case can also be worked out by checking that

the fusion rules lead to all possible L values, so for every operator in the (c, c) ring of the

model there is a corresponding chiral open string operator.

5.1. A boundary states

The intersection matrix (4.17) for the A-type boundary states with Lj = 0 is

IA = (1− g41)(1− g42)(1− g43)(1− g44)(1− g1g2g3g4). (5.3)

To determine the rank of the intersection matrix we can count the number of nonzero

eigenvalues. The gj can be diagonalized as gj = diag(1, e2πi5 , e

4πi5 , e

6πi5 , e

8πi5 ). Zero eigen-

values appear if a gj = 1 or if g1g2g3g4 = 1. The combinatorics leads to 204 nonzero

eigenvalues, which is the number of independent 3-cycles on the quintic. Thus, the Lj = 0

states provide a basis for the charge lattice. So far as we can tell they do not provide an

integral basis of the charge lattice. Furthermore, the charges of the other A-type Gepner

boundary states can be obtained from these by successive multiplication by g12

j (1+g4j ); for

example, Q(g12

1 |10000〉A) = Q(|00000〉A) + Q(g1|00000〉A), so these are even farther froman integral basis.

The intersection matrix for the |11111〉A states,

5∏

i=1

(1 + gi − g3i − g4i )

coincides with the intersection matrix (2.22) for the three-cycles Imωjzj = 0, and thus we

identify these states with the IRP 3’s.

This leads to a potential contradiction with the large volume limit in that the L = 1

states have one marginal operator, while the IRP 3’s do not. Although it might be that

this is indeed a contradiction, from what we know at present an equally likely resolution is

32

that the L = 1 marginal operator is not strictly marginal; in other words the world-volume

theory has a superpotential for the corresponding field ψ, perhaps of the form

W = ψ3 + ψφ

where φ is the Kähler modulus (ψ5 in the notation of section 3). Such a superpotential

has two ground states and would also fit the fact that the IRP 3 has a Z2 Wilson line in

the large volume limit.12

5.2. B boundary states

As we discussed in the previous section, the B-type boundary states at fixed Lj are

described by the single integer, M =∑Mj and the gj for different j are identified. The

intersection matrix (4.18) for L = 0 states can be written as:

IB = (1− g−1)5 = 5g − 10g2 + 10g3 − 5g4. (5.4)

We want to describe these boundary states in the Gepner basis. The Gepner intersection

form (3.7) in the same notation is:

Ig = −g + 3g2 − 3g3 + g4 . (5.5)

A linear change of basis preserving the action of Z5 can be written as a polynomial in the

operator g as well and a transformation of the form I → mImt will be I → Im(g)m(g−1).The relation

IB = (1− g)(1− g−1)Ig

provides this change of basis.

12 (Note added in v2): Actually, the two choices of Wilson line are topologically distinct bundles

so they would not be continuously connected in the large volume limit. This would suggest that

the potential should have a unique minimum. On the other hand, it can be shown that any

simply connected six-dimensional manifold X with H∗(X) torsion-free (such as the quintic CY)

has K(X) ∼= H∗(X), and thus the K theory class distinguishing the two bundles becomes trivial

when lifted to the CY. (We thank D. Freed and J. Morgan for explaining this to us.) Thus there

is no candidate for a space-time topological charge which could distinguish the two D-branes, and

it is not ruled out that transitions between the two choices of bundle are possible in the full string

theory.

33

The results of section 3 allow us to write these charges in the large volume basis. The

Gepner charge vector QG is related to the large volume charge vector Q as

Q = QGM−1 .

Thus QG = ( 0 1 −1 0 ) becomes Q = (−1 0 0 0 ) which is a pure (anti)six-brane.The other charges can be found by acting with the operator AL.

One can now compute the charges for the L 6= 0 branes by using the multiplicativerelation in (5.1). For example, we have

Q(g52h|10000〉B) = −Q(g2|00000〉B)−Q(g3|00000〉B).

Starting with M = 0 and successively applying this operation produces a subset of branes

which preserve the same supersymmetry. This can be checked by computing the central

charges using the periods at the Gepner point, which are simply the fifth roots of unity.

Thus the central charge for the L’th brane in this series is

Z(L) = (2 cosπ

5)LZ(0).

The charges in the Gepner basis charges can written in large volume basis viq eq.

(3.6). Tabulating these results and the numbers of marginal operators, we have (for the

Z5 representatives related to the six-brane)

L Q6 Q4 Q2 Q0 dim00000 −1 0 0 0 010000 2 0 5 0 411000 1 0 5 0 1111100 3 0 10 0 2411110 4 0 15 0 5011111 7 0 25 0 101

(5.6)

The simple pattern QL+1 = QL+QL−1 follows from the identity (−g2−g3)2 = 1−g2−g3.It is also easy to compute the number of marginal operators between pairs of distinct

boundary states. For example, |00000〉B and |(1 . . .)L(0 . . .)〉 have (for 1 ≤ L ≤ 5) 4, 3, 3, 4and 1 (respectively) marginal operators. Each corresponds to a chiral superfield of charge

(1,−1) and its charge conjugate (since the mutual intersection numbers are zero, none ofthese pairs has chiral spectra). The number of operators between two branes of higher L

of course depends on which Li are non-zero.

34

5.3. Comparison with geometrical results

To what extent can we compare these results with the geometrical branes and bundles

we discussed in section 2? The only clear match is the six-brane which indeed has no moduli

as expected.

Our states can plausibly be identified with vector bundles since they obey the stability

condition c2 > 0. We were not able to identify any of them with the explicit constructions

we mentioned in section 2. This may just reflect our lack of knowledge of vector bundles

on the quintic; thus we might regard our results as predictions of the existence of new

vector bundles. We should note that the numbers of marginal operators we obtained are

only upper bounds for the dimension of the moduli space as in general these theories will

have potentials.

The problematic objects are the |11000〉B branes as an object with these chargescannot be a classical line bundle. For reasons explained in section 2 we do not believe it

is a quantum bound state either, since we have found it at string tree level. There is a

piece of evidence that it is some sort of bound state of the six-brane with the two-brane

(2.19): namely, they come in the same multiplet of the discrete symmetries. Like all B

branes, the |11000〉B branes are invariant under Z45 , while S5 acts by permuting the Lilabels. The two-brane construction (2.19) also picks out two of the five coordinates and

thus comes in the same multiplet. This identification creates a puzzle opposite to the one

we faced for the IRIP3’s: the geometric object appears to have more moduli (12) than

the boundary state. Such a mismatch could not be fixed by a superpotential. On the

other hand, it could be that the (unknown) mechanism which binds the two-brane to the

six-brane removes moduli, so this is not a clear disagreement.

One candidate for such a bound state is the instanton in noncommutative U(1) gauge

theory [88]. Again by analogy with flat space, (since noncommutative gauge theory has not

been formulated on curved spaces, this is all we can say), at generic values of B we might

expect the D6-brane gauge theory to be noncommutative [89,90,91]. The center-of-mass

position of the instanton would then (presumably) give the moduli of a two-brane and

provide at least some of the moduli we observe. A potential problem with this idea is that

we can continue to B = 0 in the large volume limit, and there is no sign that this bound

state is unstable there.

One may ask why the D0-brane does not appear on our list. One possible explanation

is that the path from the large volume limit to the Gepner point crosses a line of marginal

35

stability, and the D0 does not exist at the Gepner point. To test this we found the periods

for all the branes in (5.6) by numerically integrating the Picard-Fuchs equations along the

negative real ψ axis. We found that the D0 is lighter than any brane from the list along the

whole trajectory, so we have no evidence for instability. Our favored explanation is simply

that all of the B branes by construction are invariant under the Z45 discrete symmetry,

while any location we might pick for the D0 would break some of this symmetry. Thus,

even if the D0 exists at the Gepner point, it cannot be a rational boundary state, at least

in this model.

6. Superpotential and topological sigma models

The calculations of the previous section describe the field content of the D-brane

world-volumes, but not their dynamics. The primary question in this regard is to find the

world-volume potential and true moduli spaces for the brane theories. In CFT language,

the marginal boundary operators operators we found might not be strictly marginal.

N = 1, d = 4 supersymmetry tells us that the world-volume potential will be a sum ofF-terms and Fayet-Iliopoulos D-terms. The D-terms are simply determined by the gauge

group and charges of the matter fields. In the case of a single brane or N identical branes

we have checked in the models we are studying that the gauge group is U(N) with all

matter uncharged under the diagonal U(1), so there is no possibility for a D-term. More

generally we must consider such terms, for example in the case of D0-branes near orbifold

points.

However, we may expect a non-vanishing superpotential, in general constrained only

by holomorphy and the symmetries of the problem. These conditions are often stronger

than they might appear, but in general the superpotential must be found by explicit

computation. It should eventually be possible to do exact calculations at the Gepner

point, as we will discuss in the next section. In this section we will try to make some

general statements about the superpotential in these models by showing that they can be

calculated as amplitudes in some topologically twisted version of the open string theory.

In particular we will use this fact to describe the cubic term in the superpotential, and to

discuss to what extent the superpotential couples to the background CY geometry.

36

6.1. Known examples of brane superpotentials

In order to motivate the search for superpotentials in these theories we will start with

a few examples where we know they arise. The most obvious example is N D3-branes in

flat space; one may write the N = 4 Lagrangian in N = 1 notation so that there are 3

adjoint complex scalar fields Zi=1,2,3 = Ziata with the superpotential Tr Z1[Z2, Z3] (here

ta are adjoint matrices for U(N)). Of course this vanishes for N = 1 but not for N > 1.

A plausible generalization of this to weak curvature (still preserving N = 1 world-

volume SUSY) is a function W written as a single trace of the adjoint chiral superfields

and with the property that

δ

δZia

δ

δZjb

δ